Uniqueness of Nonnegative Tensor Approximations

We show that a best nonnegative rank-$r$ approximation of a nonnegative tensor is almost always unique and that nonnegative tensors with nonunique best nonnegative rank-r approximation form a semialgebraic set contained in an algebraic hypersurface. We then establish a singular vector variant of the Perron--Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rank-r approximation of a positive tensor can almost never be obtained by deflation. We show the subset of real tensors which admit more than one best rank one approximations is a hypersurface, and give a polynomial equation to ensure a tensor without satisfying this equation to have a unique best rank one approximation.